 
Crystallization > Nucleation
 
Formulas for numbers of molecules on the surface of a large nucleus.by
Leonid Sakharov
Numerical simulation of nucleation phenomena during
crystallization demands a definition several parameters of the clusters of
crystalline molecules in initial amorphous phase. Among of such key parameters
are a numbers of molecules on the surface of cluster these can be transformed
from one phase to other changing number of molecules in the cluster. Lets cluster has i molecules with volume v_{m}
each. Effective size of d is defined by equation v_{m} = d^{3} The volume of the cluster is: V[i] = (4π/3) * r^{3} = i * v_{m} where r is effective radius of the cluster. The volumes of surface areas where molecules can move from
crystal into feeding environment V_{s} or into cluster V_{s}
from surrounding the nucleus phase are defined by formulas:
V_{s} = (4π/3) * (r^{3} (r g*d)^{3})
V_{s+} = (4π/3)
* ((r + g*d)^{3}  r^{3})
where g is geometrical coefficient that has meaning of
effective thickness of surface layer measured in size of molecule. Let also
introduce a constant p = 4π/3 =
4.188790205 that will help to simplify an equations. Number of molecules on surface of the cluster can be
calculated by expressions:
N_{s} = V_{s}/v_{m} N_{s+} = V_{s+}/v_{m}
Following are series of mathematical transformations to present N_{s} and N_{s+} in most convenient for calculation form:
N_{s} = (p/v_{m}) * ( r^{3} r^{3} + 3 r^{2}*gd 3 r*g^{2}d^{2} +g^{3}d^{3}) N_{s+} = (p/v_{m}) * (r^{3} + 3 r^{2}*gd
+ 3 r*g^{2}d^{2} +g^{3}d^{3}  r^{3})
Defining radius of nucleus from number of molecules:
r = p^{1/3} * i^{1/3}*v_{m}^{1/3}
and substituting it in equation for number molecules on
surface of nucleus the following formula appears:
N_{s} = (p/v_{m}) * (3 p^{2/3} * i^{2/3}*v_{m}^{2/3}*g*v_{m}^{1/3} 3 p^{1/3} * i^{1/3}*v_{m}^{2/3}*g^{2} v_{m}^{2/3} +g^{3} v_{m}) N_{s+} = (p/v_{m}) * (3 p^{2/3} *
i^{2/3}*v_{m}^{2/3}*g*v_{m}^{1/3} + 3 p^{1/3}
* i^{1/3}*v_{m}^{2/3}*g^{2} v_{m}^{2/3}
+g^{3} v_{m})
And finally getting equations:
N_{s} = 3 p^{1/3} * i^{2/3}*g 3 p^{2/3} * i^{1/3}*g^{2} +g^{3} N_{s+} = 3 p^{1/3} * i^{2/3}*g + 3
p^{2/3} * i^{1/3}*g^{2} +g^{3}
Taking that the value of geometrical coefficient could be safely estimated having value 0.85 formulas above can be presented as
N_{s} = 4.1*i^{2/3} 5.6*i^{1/3} + 0.6 N_{s+} = 4.1*i^{2/3} + 5.6*i^{1/3}
+ 0.6
It is important to point out on the remarkable fact that
formulas above do not include a volume of molecule.
Next figure shows numbers of molecules on surface of
cluster calculated by formulas above.
There is significant difference between number molecules in
the side of cluster and outside it these belong to amorphous phase. This
difference is specially is pronounced for small clusters creating pure
geometrical advantage for agglomeration of molecules. Submitted at Aug. 18, 2010; 15:16 

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